Chapter 15

Oscillatory Motion EXAMPLES

Chapter 15 Oscillatory Motion: EXAMPLES

Example 15-1: Motion of the Block (Horizontal)

The block continues to oscillate between –A and +A

These are turning points of the motion

The force F is conservative

In the absence of friction, the motion will continue forever

Real systems are generally subject to friction, so they do not actually oscillate forever

Example 15-1: Motion of the Block (Vertical)

When the block is hung from a vertical spring, its weight will cause the spring to stretch

Figure (a): Free spring, hang vertically

Figure (b): Mass m attached to spring in new equilibrium position, which occurs when:

F = mg – kxo xo = mg/k

If the resting position of the spring is defined as x = 0, the same analysis as was done with the horizontal spring will apply to the vertical spring-mass system

Which of the following represent an object undergoing SHM

(a). F = – 0.5x2 (b). F = –2.3y (c). F = 8.6x (d). F = – 4θ

Answer: (b). F = – 2.3y & (d). F = – 4θ

Force has minus sign required to restore the system to equilibrium.

Force is proportional to a displacement (which need not to be x).

Example 15-2: Conceptual Example(SHM Forces)

15-3: SHM Example 1

Initial conditions at t = 0 :

x (0)= A

v (0) = 0

This means f = 0

The velocity reaches extremes of : ± ωA

The acceleration reaches extremes of: ± ω2A

15-3: SHM Example 2

Initial conditions at t = 0 :

x (0)=0

v (0) = vi

This means f = - π/2

The graph is shifted one-quarter cycle to the right compared to the graph of x (0) = A

Suppose the spring from figure is stretched twice as far (to x = 2A). What happened to: E, vmax, & amax?

(a). E = ½kA2 (15.21)(b). vmax = ωA = (k/m)½A (15.17)(c). amax = ω2A = (k/m)A (15.18) Answers:(a). E = ½k(2A)2 = 4( kA2)

So stretching A twice as far, quadruples the energy(b). vmax = ω(2A) = 2ωA

So stretching A twice as far, doubles the maximum velocity (c). amax = ω2(2A) = 2ω2A

So stretching A twice as far, doubles the maximum acceleration

Example 15-4: Conceptual Example (Doubling Amplitude)

Material from the book to Study!!!

Objective Questions: 12-14

Conceptual Questions: 3-5

Problems: 5-8-14-18-21-25

Material for the Midterm